Revision as of 15:14, 7 December 2010

The Wheels

The Wheels can be installed in the Enigma in any sequence. If there are five different Wheels in total and the Enigma uses three Wheels. The total number of Wheel combinatins would be <math>5*4*3=60</math>

The actual mechanical Roters in a World War II Enigma. Notice Route Reflector B is installed	The scrambling action of the Enigma rotors are shown for two consecutive letters.

The software enigma

In the software Enigma you would need some random Wheels and a Random Rotor Reftlector. For examle this three bit Enigma.

Three bit Example

Crypt

In the example below if the input=2 its send to Wheel 1 position 2 which output will be 1 and that's input for Wheel 2 which outputs 0 to the Reflector that outputs 7 as input for Reverse Wheel 2 which output's 2 as input for Reverse wheel 1 which Output 0 as the encrypted message.

2->1->0->7->2->0

Input                      0,   1,   <notice>2</notice>,   3,   4,   5,   6,   7 
Wheel 1                    2,   7,   <notice>1</notice>,   5,   6,   3,   0,   4
Wheel 2                    5,   <notice>0</notice>,   7,   2,   1,   6,   4,   3
Reflector                  <notice>7</notice>,   4,   6,   5,   1,   3,   2,   0
Reverse wheel 2            1,   4,   3,   7,   6,   0,   5,   <notice>2</notice>
Reverse wheel 1(output)    6,   2,   <notice>0</notice>,   5,   7,   3,   4,   1

Decrypt

Reverse: 0->2->7->0->1->2

Input                      <notice>0</notice>,   1,   2,   3,   4,   5,   6,   7 
Wheel 1                    <notice>2</notice>,   7,   1,   5,   6,   3,   0,   4
Wheel 2                    5,   0,   <notice>7</notice>,   2,   1,   6,   4,   3
Reflector                  7,   4,   6,   5,   1,   3,   2,   <notice>0</notice>
Reverse wheel 2            <notice>1</notice>,   4,   3,   7,   6,   0,   5,   2
Reverse wheel 1(output)    6,   <notice>2</notice>,   0,   5,   7,   3,   4,   1

The wheel notch

Each time a letter/digit is encrypted/decrypted - same process - the Wheels are ticked forward.

Tick 1

Wheel 1 is ticked forward on every encryption/decryption. See below. When the Notch - show in red below - ticks from 7 to 0 it will Tick the next Wheel. See Tick 2.

Wheel positions before Tick number 1

Input                      0,   1,   2,   3,   4,   5,   6,   7 
Wheel 1                    <notice>2,   7,   1,   5,   6,   3,   <error>0</error>,   4</notice>
Wheel 2                    5,   0,   7,   2,   1,   6,   <error>4,</error>   3
Reflector                  7,   4,   6,   5,   1,   3,   2,   0
Reverse wheel 2            <notice>1,   4,   3,   7,   6,   0,   5,   2</notice>
Reverse wheel 1(output)    6,   2,   0,   5,   7,   3,   4,   1

Wheel positions after Tick number 1

Input                      0,   1,   2,   3,   4,   5,   6,   7 
Wheel 1                    <notice>4,   2,   7,   1,   5,   6,   3,   <error>0</error></notice>
Wheel 2                    5,   0,   7,   2,   1,   6,   <error>4</error>,   3
Reflector                  7,   4,   6,   5,   1,   3,   2,   0
Reverse wheel 2            1,   4,   3,   7,   6,   0,   5,   2
Reverse wheel 1(output)    <notice>2,   0,   5,   7,   3,   4,   1,   6</notice>

Tick 2

Wheel 1's Notch is ticked from 7 to 0 it will Tick wheel 2 forward. The reflector will stay stationary.

Wheel positions before Tick number 2

Input                      0,   1,   2,   3,   4,   5,   6,   7 
Wheel 1                    <notice>4,   2,   7,   1,   5,   6,   3,   <error>0</error></notice>
Wheel 2                    5,   0,   7,   2,   1,   6,   <error>4</error>,   3
Reflector                  7,   4,   6,   5,   1,   3,   2,   0
Reverse wheel 2            1,   4,   3,   7,   6,   0,   5,   2
Reverse wheel 1(output)    <notice>2,   0,   5,   7,   3,   4,   1,   6</notice>

Wheel positions after Tick number 2

Input                      0,   1,   2,   3,   4,   5,   6,   7 
Wheel 1                    <notice><error>0</error>,   4,   2,   7,   1,   5,   6,   3</notice>
Wheel 2                    <notice>3,   5,   0,   7,   2,   1,   6,   <error>4</error></notice>
Reflector                  7,   4,   6,   5,   1,   3,   2,   0
Reverse wheel 2            <notice>2,   1,   4,   3,   7,   6,   0,   5</notice>
Reverse wheel 1(output)    <notice>6,  2,   0,   5,   7,   3,   4,   1</notice>

Crypto strength of three bit Enigma

If the Wheels and Reflector are known to the hacker

If there are to total of five wheels of which to are installed. That would give <math>5 * 4 = 20</math> wheel combinations.
Each installed wheel can be in one of eight start positions. That would give <math>8 * 8 = 64</math> start combinations.
Total combinations if the five wheels are known <math>20 * 64 = 1280</math> combinations.

If the Wheels and Reflector are unknown to the hacker

A three bit wheel has <math>!8 = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320</math> combinations.
There are five Wheels <math> 5 * 40320 = 201600</math>
The Reflecter which is symmetrical has <math>!7 = 5040</math> combiantions.
Total combinations with unknown wheels <math>201600 * 5040 * 1280 = 1.300.561.920.000</math>

Not that impressive, see the 8 bit example below

8 bit example

Number of possible keys when using three wheels

In the example below there is a caculation of possible keys that is

Three whells of 8 bit each diffently coded (fx. wheel[0]=67,wheel[1]=234 .. wheel[67]=165 .. )
An 8 bit Rotor-Reflector which is symmetricaly coded. (Fx. rr[0]=67 then rr[67]=0, rr[87]=167 then rr[167]=87 etc. )
When the encryption/decryption starts each wheel can be in 1 of 256 positions. Called the notch position

heth@MachoGPU:~/enigma$ <notice>bc</notice>
bc 1.06.95
Copyright 1991-1994, 1997, 1998, 2000, 2004, 2006 Free Software Foundation, Inc.
This is free software with ABSOLUTELY NO WARRANTY.
For details type `warranty'.
<notice>/* Define the Factorial function */ 
define frac(x) {
  if (x>1) {
    return (x * frac (x-1))
  }
  return (1)

}

/* Number of possible alterations of a 8 bit wheel */
wheel=frac(256)
print wheel</notice>
85781777534284265411908227168123262515778152027948561985965565037726\
94525531475893774402913604514084503758853423365843061571968346936964\
75322289288497426025679637332563368786442675207626794560187968867971\
52114330770207752664645146470918732610083287632570281898077367178145\
41702505230186084953190681382574810702528175594594769870346657127381\
39286205234756808218860701203611083152093501947437109101726968262861\
60626366243502284094419140842461593600000000000000000000000000000000\
0000000000000000000000000000000

<notice> /* Number of installed wheels in the Enigma engine*/
numberofwheels=3

/* Number of possible alterations of the 8 bit rotor-reflector.
   The rotor-reflector can't point to it-self. (fx. location 8 can't be 8)
*/
rotorreflector=frac(255)
print rotorreflector</notice>
33508506849329791176526651237548149420225840635917407025767798842862\
08799035732771005626138126763314259280802118502282445926550135522251\
85672769253319307041281108333032565932204170002979216625073425339051\
37544660457112403384627010340202629925813784231472766366436471553963\
05352541105541439434840109915068285430675068591638581980604162940383\
35658673919826878210492461407660579356286524198217620742862096977680\
31494674313868079724382476891586560000000000000000000000000000000000\
00000000000000000000000000000
<notice>

/* Number of possible startpositions of the three wheels.
   they are called a notch from the original mechanical solution
*/ 
notches=256*256*256
print notches</notice>
16777216
<notice>
/*Total number of Keys when using a 8 bit Enigma with three wheels */
wheel*numberofwheels*rotorreflector*notches</notice>
14467425940815419878550914285328747194412838285887202913906796612290\
67116765686074239837002699523310587812021105128921519898929662849673\
80231859551685345801700678590340949512766773530970639425397581454798\
34629363249827620200556239557412599222204518999540726185192515996315\
06978129392323972906002241190936561257411009104030200763332968508802\
10053380885667825490547810051729838486330141341759611086561117956500\
02210195590742957751653052758866489611046357269440002628221514248948\
17132504901094529220429149272922120473850361954385338800798632554213\
60438811591136370963675381915493290515801660130577959111839885736840\
77541504957308491144024012326090397003948528853785364705857026051935\
47050340165465982389653696093503539510788359067272108792866788729818\
77493077624052679059588612553113715864219395902995530561734847463169\
09861780870355495760296743612825305064871402809304345105972352826182\
65297223680000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000

If we used a 10 bit rotor

Or try and imagine a 32 bit wheel (Then we would need a better implementation of frac)

<notice>frac(1024)</notice>
54185287960588572830769219446838547380015539635380134444828702706832\
10612073376603733140984136214586719079188457089807539319941657701873\
68260454133333721939108367528012764993769768292516937891165755680659\
66374794731451840488667767255612518869433525121367727452196343077013\
37132057962484331288700884361716546902375183904529447322778084029321\
58722061853806162806063925435310822186848239287130261690914211362251\
14468471388858788162925210404629531594994390035788241024393431503744\
41138908061814062108639532752353758850185984515822295996545585412427\
89130902486944298610923153307579131675745146436304024890820442907734\
56182736903050225279692655307296737099075874779312763510470246988966\
79614621330262371589732278578146318071564277676440645910850765647834\
56324457736853810336981776080498707767046394272605341416779125697733\
37456803747518667626596166561588468145026333704252266414186215704682\
56847733609443267374936766749150989537681129458316266438564790278163\
85730291542667725665642276826058264393884514911976419675509290208592\
71315636298329098944105273212518724952750131407167640551693619078182\
12367019122957673631170541265899299164820085157817519554669109028387\
29232224509906388638147771255227782631322385756948819393658889908993\
67087451686065309841102029985381628156433498184710577783953474253149\
96221034888075845137057698397639931039296650460461211666513451311495\
13657400869056334867859885025601787284982567787314407216524272262997\
31979156860362940662474010148269755953315573665880056292127468065728\
52015704019406922855578006114290557553245497940089398491468126398607\
50085263298820224719585505344773711590656682821041417265040658600683\
84494510435499881288680131655155171467338832334085176381971359131237\
25486737347835373163415173693875652128997265979649032412087273486906\
99802996369265070088758384854547542272771024255049902319275830918157\
44820519642107283720493729351617534195777542245315244228039137240771\
78916612030610402558300550338867900521160254087404546209383843676378\
86658769912790922323717371343176067483352513629123362885893627132294\
18356588401041872786935443907708527828855830842709046107501900718493\
31399155582127523923298797806496390753338457191738228405018695704636\
26600235265587502335595489311637509380219119860471335771652403999403\
29636024557725796367328665434895732574099971056713162327234576676193\
76514081039991936339082864205100985774545240681068973924931382873622\
26257920000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000000000

When the wheels are known

Like during World War II the Allied get hold of the Wheels for the Enigma and only had to find the order of the Wheels and the startposiotion of each installed wheel.

So with the 8 bit Enigma engine with three wheels installed. There are 10 known wheels in the example below.

<notice>/* Total number of wheels possible */
wheelstotal=10

/* With three wheels installed there would be 10*9*8 ways configure */
wheelcombinations=10*9*8

/* Each wheel can start in 256 ways */
notches=256*256*256

/* Total number of keys, when the wheels are known */
wheelstotal*wheelcombinations*notches</notice>
120795955200

@@ Line 121: / Line 121: @@
 define frac(x) {
    if (x>1) {
-     return (x * f (x-1))
+     return (x * frac (x-1))
    }
    return (1)
@@ Line 183: / Line 183: @@
 </source>
 |}
 === If we used a 10 bit rotor ===
 Or try and imagine a 32 bit wheel (Then we would need a better implementation of ''frac'')

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